Maths · Binomial theorem for a positive integral index
The coeffcient of in the expansion of ^{2}\left(1+x^{2}\right)^{3}\left(1+x^{3}\
The coeffcient of \( x^{10} \) in the expansion of \( (1+x)^{2}\left(1+x^{2}\right)^{3}\left(1+x^{3}\right)^{4} \) is equal to
- A. 52
- B. 44
- C. 50
- D. 56
Step-by-step solution
Expand each factor: (1+x)^2 = Σ C(2,a) x^a, (1+x^2)^3 = Σ C(3,b) x^{2b}, (1+x^3)^4 = Σ C(4,c) x^{3c}. Need a+2b+3c=10 with a∈{0,1,2}, b∈{0,1,2,3}, c∈{0,1,2,3,4}. Valid triples: (1,3,1): 2*1*4=8; (0,2,2): 1*3*6=18; (2,1,2): 1*3*6=18; (1,0,3): 2*1*4=8. Sum = 52.